Supply of Insurance:

1
Economics of Insurance 1
Lecture 3
Supply of Insurance What do insurance companies
provide?
2
What insurance companies do Some key words
Insurance companies REDUCE the financial RISK to
their clients arising from BAD (insured
against) EVENTS.
Financial risk (risk in the following slides)
is TRANSFERRED from the client to the insurance
company.
An insurance company and a client make the
transfer through a CONTRACT
The deal is that for a PREMIUM paid by the client
to the insurance company, the client receives
COMPENSATION if the bad event occurs.
The total amount of compensation the company has
agreed to pay-out if the bad event occurs is
called the COVER
When a client experiences a event that she has
insured against they make a CLAIM to the
insurance company for compensation.
The claim will be for compensation to cover all
or some of the financial LOSS the client has made
as a consequence of the bad event
3
More words and concepts
But this appears to make no sense How can a
company be any more successful in dealing with
risk than any of its individual clients? Why
isnt the risk the company faced simply
multiplied by the number of clients it has got?
Its success is because the insurance company (the
company in the following slides) POOLS the risk
of each of the MANY individual clients.
And, in one way or another this pooling REDUCES
the risk the company takes compared to each
individual client.
In this respect the company is more EFFICIENT in
dealing with the risk than any of its individual
clients and so it can trade with each to mutual
advantage.
4
An intuitive view of why pooling works
10 houseowners each own a house worth 100000
undamaged plus another 30000 income
Each house has 10 chance of burning down, value
becoming 0
10 houseowners contribute to a mutual insurance
pool
Club rule Each owner pays 10000 (10 - the
probability of fire - of 100000) into the pool
which will COMPENSATE any houseowner whose house
is burned down
Houses
and asset values
120k
130k
Pool has a value of
120k
130k
120k
130k
10k
10k
10k
120k
Insurance Pool
130k
120k
130k
20k
120k
130k
10k
10k
130k
120k
100000
10k
10k
10k
10k
120k
130k
10k
120k
130k
On average 1 house burns down,, each year
120k
130k
and we can expect 9 houseowners to have undamaged
houses in any year
So there will just be enough money in the pool to
compensate the damaged-house owner,
and the damaged-house owner will make no
FINANCIAL loss because of the fire.
5
AVOIDING RISK BY AVERAGING IN A POOL
6
Averaging Risks
AVERAGING RISKS
The previous intuitive view of pooling has many
characteristics which we shall discuss in this
lecture.
The FIRST EFFECT is based on AVERAGING the risks
of individuals.
We dealt with this intuitively on a previous
slide in which our householders agreed to
contribute to a mutually owned pool of resources
that would compensate any one of them whose house
burned down.
With each house owners contributing a tenth of
the value of their house each year and a 10
chance of the any house actually being destroyed
by fire, the mutual fund would be just enough to
compensate one person for the loss of their
house, and we would expect one person each year
to have their house being burned down.
ON AVERAGE there would be one fire, causing
100000 worth of damage, compensated through the
pooled fund of 100000 just big enough to pay
the compensation.
ON AVERAGE therefore, the pool carries NO RISK
there is just enough money to pay agreed
compensation.
7
Pooling and Averaging Risks
AVERAGING RISKS
An insurance firm is an organisation that buys
(thereby TRANSFERS) risk from individual clients
to itself and organises those risks into the POOL
So it reduces risks (as we have just seen) ON
AVERAGE to zero
So this is the first sense in which the company
is dealing with risks more efficiently than each
individual client
Lets see how this works with a simple example
8
Averages and Expectations
AVERAGING RISKS
The following are the number of claims an
insurance firm has received for each of the last
9 years
7000
5000
8000
7000
4000
6000
6000
6000
5000
The AVERAGE (arithmetic mean) number of claims is
6000
We could call the average the EXPECTED NUMBER of
claims. When we use the word EXPECTED, it means a
kind of AVERAGE as shown above. AN EXPECTATION is
often written as E() where the contents of the
bracket is the THING we are expecting in this
case number of claims.
That is to say E(number of claims) 6000
9
Fair premium rates always cover expected
compensation costs
If a company charges a FAIR premium
it will have just enough income to pay out the
expected number of claims
7000
5000
8000
7000
4000
6000
6000
6000
5000
In our example
E(number of claims) 6000
Say the probability of making a claim is 0.05
This means that number of clients is 6000/0.05
120000
and the FAIR gross premium rate is 0.05 ( 5 pence
per ) see Lecture 2
Say each claim is for 2000.
The FAIR premium must be 0.05 x 2000 100
So total premium income is 120000 x 100 12m
AND
Total compensation payments 6000 x 2000 12m
CHARGING A FAIR PREMIUM RATE ALWAYS CREATES JUST
ENOUGH INSURANCE COMPANY INCOME TO PAY THE
EXPECTED NUMBER OF CLAIMS
10
Expectations are usually wrong!
AVERAGING RISKS
The average of these numbers is 6000 the
insurance company EXPECTS 6000 claims each year.
7000
5000
8000
7000
4000
6000
6000
6000
5000
But that expectation was WRONG 6 years out of 9!
A serious number of mistakes for the insurance
company
Especially when there were more claims than
expected !
But there are things that can done about this
these raise the second characteristic of risk
pooling in insurance
11
ALLOWING FOR ANNUAL VARIATION IN CLAIM NUMBERS
12
Frequency of annual claim numbers
ALLOWING FOR ANNUAL VARIATION IN CLAIM NUMBERS
Lets count how many times each number occurs
5
8
7
4
7
6
6
6
5
Now lets plot these results on a graph
0
1
1
2
3
1
2
1
2
1
0
13
Probability distribution
ALLOWING FOR ANNUAL VARIATION IN CLAIM NUMBERS
If the insurance company knows the probability
distribution function it can make provision years
of higher than average claim number
But we need a few more pieces of the distribution
analysis before looking at how the company can
manage the risk.
This gives us a PROBABILITY DISTRIBUTION FUNCTION
0 0.11111 0.222222 0.33333
Probability
No of Claims
1 2 3 4 5 6
7 8 9 10 11
Mean
14
Interpretation of the probability diagram 2
ALLOWING FOR ANNUAL VARIATION IN CLAIM NUMBERS
Now lets put the information about probabilities
and claim numbers into a table
We could say that 4000 represents annual claim
numbers between 3500 and 4500.
probability
(4.5-3.5)
x (0.11111 -0)
0.11111
(5.5-4.5)
x (0.22222 -0)
0.22222
x (0.33333 -0)
(6.5-5.5)
0.33333
So we could change our point coordinate
x (0.22222 -0)
0.22222
(7.5-6.5)
(8.5-7.5)
x (0.11111 -0)
0.11111
Into the column of a histogram
Whose area we can calculate
0 0.11111 0.222222 0.33333
Probability
0.33333
0.33333
0.22222
0.22222
0.22222
0.11111
0.11111
0.11111
0.11111
No of Claims
1 2 3 4 5 6
7 8 9 10 11
Mean
1
1
1
1
1
15
Probability density
ALLOWING FOR ANNUAL VARIATION IN CLAIM NUMBERS
probability
probability density
(4.5-3.5)
x (0.11111 -0)
0.11111
(5.5-4.5)
x (0.22222 -0)
0.22222
x (0.33333 -0)
(6.5-5.5)
0.33333
x (0.22222 -0)
0.22222
(7.5-6.5)
(8.5-7.5)
x (0.11111 -0)
0.11111
probability density
This is often called
On a semantic note lets consider the name we have
attached to the vertical variable probability.
and the mathematical form of the graph such as
our one on the diagram a
Actually its the probability of a single column
in the diagram.
Probability density
0 0.11111 0.222222 0.33333
e.g. the probability of the second column (the
number of claims between 4500 and 5500) is
0.22222.
Probability
probability density function
So we should really label the probability axis
No of Claims
probability per each group of 1000 claims
1 2 3 4 5 6
7 8 9 10 11
Mean
16
Interpretation of the probability diagram
cumulative area
ALLOWING FOR ANNUAL VARIATION IN CLAIM NUMBERS
probability density
(4.5-3.5)
x (0.11111 -0)
0.11111
0.11111
(5.5-4.5)
x (0.22222 -0)
0.22222
0.33333
x (0.33333 -0)
(6.5-5.5)
0.33333
0.66666
x (0.22222 -0)
0.22222
(7.5-6.5)
0.88888
(8.5-7.5)
x (0.11111 -0)
0.11111
0.99999
Notice that the cumulative area of all the
rectangular columns is equal to 1
ADDING THE AREA OF EACH RECTANGLE TOGETHER
probability density
GIVES THE CUMULATIVE AREA OF THE RECTANGLES
0 0.11111 0.222222 0.33333
0.66666
0.33333
0.88888
1.00000
0.11111
No of Claims
1 2 3 4 5 6
7 8 9 10 11
Mean
17
Area of rectangles and area under the line
ALLOWING FOR ANNUAL VARIATION IN CLAIM NUMBERS
probability density
(4.5-3.5)
x (0.11111 -0)
0.11111
0.11111
(5.5-4.5)
x (0.22222 -0)
0.22222
0.33333
x (0.33333 -0)
(6.5-5.5)
0.33333
0.66666
x (0.22222 -0)
0.22222
(7.5-6.5)
0.88888
(8.5-7.5)
x (0.11111 -0)
0.11111
0.99999
LOOK AT THE AREA OF EACH RECTANGLE AND COMPARE IT
TO THE AREA UNDER THE LINE (area under the
function)
SO WE CAN SAY THAT THE AREA UNDER THE SMOOTH
FUNCTION LINE IS SIMILAR TO THE AREA FOR ITS
EQUIVALENT RECTANGLE
probability density
e.g. column 2 has an area 0f 0.22222
0 0.11111 0.222222 0.33333
0.33333
0.22222
0.22222
Which is identical to the area under the line for
the same interval
0.11111
0.11111
No of Claims
1 2 3 4 5 6
7 8 9 10 11
Mean
18
Interpretation of the probability diagram
cumulative probability
ALLOWING FOR ANNUAL VARIATION IN CLAIM NUMBERS
probability density
cumulative probability
0.11111
(4.5-3.5)
x (0.11111 -0)
0.11111
0.11111
0.33333
(5.5-4.5)
x (0.22222 -0)
0.22222
0.33333
0.66666
x (0.33333 -0)
(6.5-5.5)
0.33333
0.66666
x (0.22222 -0)
0.22222
(7.5-6.5)
0.88888
0.88888
(8.5-7.5)
x (0.11111 -0)
0.11111
0.99999
1.00000
Probability of no more than 4000 claims in any
year
Certainty of 8000 or fewer claims in any year
Probability of 6000 or fewer claims in any year
Probability of 7000 or fewer claims in any year
Probability of 5000 or fewer claims in any year
REMEMBER THAT AREAS REPRESENT PROBABILITIES
The cumulative area of all the rectangles in the
diagram comes to 1.
probability density
0 0.11111 0.222222 0.33333
So the cumulative probability is ONE
So cumulative areas represent CUMULATIVE
PROBABILITIES
0.66666
0.33333
0.88888
1.00000
0.11111
No of Claims
1 2 3 4 5 6
7 8 9 10 11
Mean
19
How an insurance company could manage risk with
this model
ALLOWING FOR ANNUAL VARIATION IN CLAIM NUMBERS
probability density
(4.5-3.5)
x (0.11111 -0)
0.11111
0.11111
0.11111
(5.5-4.5)
x (0.22222 -0)
0.22222
0.33333
0.33333
x (0.33333 -0)
(6.5-5.5)
0.33333
0.66666
0.66666
x (0.22222 -0)
0.22222
(7.5-6.5)
0.88888
0.88888
(8.5-7.5)
x (0.11111 -0)
0.11111
0.99999
1.00000
Then the company could use the model to quantify
the risk, and thus make a managerial decision how
much risk to take
If this was a correct model of the risk that an
insurance company might face a higher than
average number of claims in any year
probability density
0 0.11111 0.222222 0.33333
No of Claims
1 2 3 4 5 6
7 8 9 10 11
Mean
20
Managing the risk of an excess number of claims
ALLOWING FOR ANNUAL VARIATION IN CLAIM NUMBERS
The insurance company might feel that it was
prepared keep reserve funds available so that it
could pay compensation to clients 89 years in
every 100.
i.e. it was prepared for 88.888 of years
On our diagram the 88.888 risk takes us to the
fourth column
That is it must have enough reserves to pay out
7000 claims each year
Assuming it was receiving enough from premiums to
cover the average number of claims (6000) per year
It must have enough reserve funds to pay up a
thousand extra claims
probability density
0 0.11111 0.222222 0.33333
0.88888
No of Claims
1 2 3 4 5 6
7 8 9 10 11
Mean

21
A Normal Distribution
ALLOWING FOR ANNUAL VARIATION IN CLAIM NUMBERS
So the second way an insurance company can manage
risk is by using a model to estimate the probable
excess (above average) number of claims and then
keeping a large enough reserve fund to allow for
them.
But we need a more sophisticated model than the
one used so far.
Lets say our model of insurance company risk is a
bell curve
A NORMAL DISTRIBUTION
Notice that this probability density function
also has an area approximated by rectangular
columns
although the approximation is not as good as for
our previous straight line density function
probability density
0 0.11111 0.222222 0.33333
1 2 3 4 5 6
7 8 9 10 11
Mean
22
Cumulative Probability and the Normal
Distribution
ALLOWING FOR ANNUAL VARIATION IN CLAIM NUMBERS
So the area under the bell curve is the
cumulative probability of the number of claims
specified or fewer.
or the probability of there being 7000 claims or
fewer is the area under the curve with the blue
diagonals
e.g. the probability of there being 5000 claims
or fewer is the area under the curve with the red
diagonals
If we knew more about this density function then
we could measure without approximation the area
under the curve and thus the cumulative
probability of claim numbers.
Notice that this probability density function
also has an area approximated by rectangular
columns
although the approximation is not as good as for
our previous straight line density function
probability density
0 0.11111 0.222222 0.33333
So lets explore the normal distribution curve a
bit more
1 2 3 4 5 6
7 8 9 10 11
Mean
23
Properties of the normal distribution
ALLOWING FOR ANNUAL VARIATION IN CLAIM NUMBERS

• The line is CONTINUOUS. The rectangles have an
  arbitrarily small width. e.g. we might measure
  claim numbers in ONES rather than thousands
• Hence the sum total of the areas of the
  rectangles is as close as you wish it to be to
  the area under the line.
• Mathematically you are INTEGRATING each of
  the narrow rectangles to get their sum total area.

2) The area under the normal distribution line
between any two claim numbers is the probability
of claim numbers being between the two in any
year.
3) The total area beneath the normal distribution
line is ONE
4) The mean number of claims has the highest
probability. i.e. the mean corresponds to the TOP
of the distribution
6) The variations from the mean are regular and
can be understood using the concept of
probability density
Hence there is no need to show the rectangles
5) The variation of (or deviation from) claim
numbers from the mean is symmetrical. i.e. the
distribution is symmetrical about the mean
Prob that the number of. claims will be between
A and the mean
ONE
Mean
A
STANDARD DEVIATION
24
Standard Deviation
ALLOWING FOR ANNUAL VARIATION IN CLAIM NUMBERS
Standard deviation is a measure of how much the
distribution function deviates from the mean.
Note how for our normal distribution the
deviation from the mean increases as
probabilities decrease.
That is what gives the normal distribution (and
many other distributions) its bell shape
The definition of standard deviation is that it
is the root mean square of the deviations from
the mean
You may remember from your statistics that the
standard deviation is the square of the
variance
Standard deviation is a kind of average of all
the deviations from the mean
Mean
25
Deviation from mean
ALLOWING FOR ANNUAL VARIATION IN CLAIM NUMBERS
So, for example, in a particular year the
insurance company receives say 4000 claims,
that is 2000 less than the mean.
The absolute deviation from the mean for that
year is 2000
In another year the insurance company may receive
7000 claims,
that is 1000 more than the mean.
The absolute deviation from the mean for that
year is 1000
The AVERAGE deviation for the two years is 1500
If we averaged every deviation between the mean
and the probability function (ie the curve) we
would get the standard deviation
Standard deviation is a kind of average of all
the deviations from the mean
1000
2000
Mean 6000
4000
7000
26
Standard Deviation
ALLOWING FOR ANNUAL VARIATION IN CLAIM NUMBERS
Standard deviation is the average of the length
of all the animated arrows in the diagram
If we averaged every deviation between the mean
and the probability function (ie the curve) we
would get the standard deviation
Standard deviation is a kind of average of all
the deviations from the mean
Mean
Average length of animated arrow
one standard deviation
27
Comparing Different Standard Deviations
ALLOWING FOR ANNUAL VARIATION IN CLAIM NUMBERS
Here is our original distribution
with its standard deviation
As the standard deviation decreases
So with the new, orange, distribution, you would
expect actual number of claims to deviate less,
on average, from the mean than with old black
distribution
the function changes it gets closer to the mean
probability density
No of Claims
1 sd
1 sd
28
Comparing Different Standard Deviations
ALLOWING FOR ANNUAL VARIATION IN CLAIM NUMBERS
Suppose there were two insurance companies
The Black company had the black probability
distribution
The Orange company had the orange probability
distribution
With the orange, distribution, you would expect
actual number of claims to deviate less, on
average, from the mean than with black
distribution
probability density
Which company would be more at risk from a year
with exceptionally high claim numbers?
No of Claims
1 sd
1 sd
29
Comparing Risk of two Insurance Companies
ALLOWING FOR ANNUAL VARIATION IN CLAIM NUMBERS
Suppose there were two insurance companies
The Black company had the black probability
distribution
Both companies experience claim numbers sometimes
more, sometimes less than average.
But
The Orange company had the orange probability
distribution
with the orange distribution, you would expect
actual number of claims to deviate less, on
average, from the mean than with black
distribution
SO
Although Orange Co. has as many YEARS of
excessive claim numbers as the Black Co., on
average the NUMBER of claims in any excess year
for Orange will be less than the NUMBER for Black.
probability density
Which company would be more at risk from a year
with exceptionally high claim numbers?
In that sense Orange Co. experiences less risk
than Black Co.
No of Claims
1 sd
1 sd
30
2 Key Points
ALLOWING FOR ANNUAL VARIATION IN CLAIM NUMBERS
The risk to an insurance company of having more
than average claim numbers in any year is
measured, in this model, by the size of the
standard deviation (s) compared to the mean
The bigger the ratio standard deviation/mean,
the bigger the risk
just as in finance and investment theories
Insurance companies try to reduce the standard
deviation of claim numbers (and values) compared
to the mean
They try and keep their probability distribution
as close to the mean as possible
31
Standard Deviation and Risk
ALLOWING FOR ANNUAL VARIATION IN CLAIM NUMBERS
The NORMAL DISTRIBUTION is used as a model for
risk because it has a very well known
relationship between standard deviation and
probability
This relationship is represented by the z
statistic
The probability of X being between either
mean and (mean MINUS one standard deviation)
OR mean and (mean PLUS one standard deviation)
is ALWAYS
And X being between mean and 3 standard
deviations from the mean has a probability of
about 49.73
You can choose any number of standard deviations
from the mean and each will always give the same,
unique probability
pd
34.13
Similarly X being between the mean and (mean 2
sds) always occurs with a probability of about
45.45
Or you could choose a probability and that would
always correspond to a unique number of standard
deviations
0.3413
The range of z statistics, from which these
correspondences can be found, is given on
spreadsheets, statistical and finance packages,
printed tables
0.3413
0.4545
0.4545
X
(No of Claims)
32
Volatility and profitability.
Now lets see how we can use the concept of a
standard deviation to analyse insurance company
decisions
A small insurance company with say 10000 clients,
each with a probability of 0.1 of claiming 5000
in any year, will on average receive 1000 (i.e.
0.1x10000) claims each year. This is the expected
number of claims.
On this basis it expects 550000 profit
BUT suppose our company experienced a year with
1010 claims rather than the average  1000 claims.
Actual profit expected profit compensation
paid on the 10 extra claims Actual profit   
550,000                       
       50000                             
500,000
It has obviously performed worse than expected
so the firm can expect adverse consequences
33
How much of a reserve should the company hold to
avoid such an unexpected reduction in profit?
Note Holding reserves will be costly and thus
also reduce profit.
So the PROFIT IDENTITY now reads
E(P) R E(G)    A     v.V
where V is the value of the volatility fund and v
the rate of servicing the fund.
THIS MEANS THAT PROVISION FOR UNEXPECTED PROFIT
REDUCTION OR LOSS MUST BE BALANCED AGAINST THE
COST OF THAT PROVISION
No insurance company wil make infinite provision
for unexpected claim numbers
34
How much of a reserve should the company hold to
avoid such an unexpected reduction in profit?
Using the previous results, if the company knew
the standard deviation of claim numbers it could
find the probability of excess claims and balance
this against the cost of holding the reserve.
In this case the standard deviation is 30 (how
found?)
From ztables the company would know that there
was an 84.13 chance of there being 1030 or
fewer claims in any year
Assuming claims are normally distributed
35
How much of a reserve should the company hold to
avoid such an unexpected reduction in profit?
So if the company wanted to be sure it could pay
out claims in 84 years out of 100
Size of standard deviation
It would hold a fund of 30 x Gross Compensation
30 x 5000
150000
Which would cost say 10 of 150000
15000
i.e. its expected profits would be reduced by
15000
36
How much of a reserve should the company hold to
avoid such an unexpected reduction in profit?
Suppose the company wanted to be sure of being
able to pay out 99 years in 100
i.e. with a probability of 99
From z tables, a probability of 99 is found
about 3 standard deviations above the mean
Size of standard deviation
So it must make provision for 3 x 30 90
additional claims
This requires a fund of 90 x 5000 450000
Which would cost 45000
A much bigger reduction in profit
For a much bigger safety margin
37
How much of a reserve should the company hold to
avoid such an unexpected reduction in profit?
What would the insurance company do?
This is a management decision
But they do have the trade-off between profit
and risk on which to base their decision
38
But statistics is helpful in another respect
The so called Law of Large Numbers
This tells us that the larger the population, the
closer the standard deviation will be to the mean
A bit like this
probability density
No of Claims
1 sd
1 sd
39
But more like this
If the distributions are normal
Doubling the number of clients
Doubles the MEAN
1 sd
1 sd
Mean
No of Claims
Mean
The standard deviation increases,
BUT NOTHING LIKE DOUBLES
40
Recall that the insurance company with a standard
deviation RELATIVELY closer to its mean is less
risky
1 sd
1 sd
Mean
No of Claims
Mean
So the company on the black distribution is more
risky than
The company on the orange distribution
41
So Orange Company needs to make PROPORTIONATELY
less provision for excess claims
than the Black Company
Why?
BECAUSE THE ORANGE COMPANY IS BIGGER (HAS MORE
CLIENTS)
AN INSURANCE COMPANY CAN REDUCE ITS RISK BY
GROWING
This is a real economy of scale
Thus average costs can be reduced by growing
MAKING PROFIT PER CLIENT IS BIGGER
42
In Conclusion
The BASIC ways in which insurance companies
handle client risk are
POOLING clients risks
MODELLING risk and making calculated provision
for the volatility of claim number
INCREASING the number of clients